The solutions of Partial Differential Equations are extensive and calculation-demanding applications that require high performance numerical computing and fast communication channels. Grids are often not the ideal candidates for the solution of such problems. We evaluated experimentally the performances of Grid5000 in the solution of a diffusion equation using finite difference methods and MPI communication library. Measurements of the point-to-point communication indicate a significant variation in communication speed between Grid computers. There are also differences in the computational performances of the Grid computers, either because of different hardware and software resources, or because of sharing the computing power by other users. Both variations could be time dependent. It follows that a quite complex load balancing policy is needed for compensating the differences in the numerical and communication performances and for the identification of the optimal set of cooperating computers.